Grenzwert der Fibonacci-Folge, F0=0, F1=1 Fn+1=Fn+Fn− ⇔ Fn+1 Fn = A 1 1 1 0 ⋅ Fn Fn−1 Fn+1 Fn = 1 1

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    06-Mar-2018

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  • Fibonacci-Folge

    F0 = 0, F1 = 1

    Fn+1 = Fn + Fn

    Fn+1Fn

    =

    A

    1 1

    1 0

    Fn

    Fn1

    Fn+1Fn

    =

    1 1

    1 0

    n

    F1F0

    =

    1 1

    1 0

    n

    1

    0

    Berechnung der Eigenwerte: A x =

    x A

    x E

    x =

    0 (A E)

    x =

    0

    .det(A E) = 0 x =

    0

    Ist der Eigenvektor , dann mu die Determinante sein:x

    0 = 0

    1 11

    = 0 (1 ) 1 = 0 + 2 1 = 0 2 + 14= 54 1

    2= 12

    5

    =1+ 5

    2 =

    1 52

    Berechnung der Eigenvektoren:

    1 11

    x

    y

    =

    0

    0

    (1 )x + y = 0x y = 0

    y = ( 1)xx = y

    :

    x

    y

    =

    yy

    = y

    1

    =

    1+ 52

    x =

    1+ 52

    1

    ; =

    1 52

    x =

    1 52

    1

    Explizite Form der Fibonacci-Folge:

    Darstellung des Vektors als LK der Eigenvektoren:

    1

    0

    1

    0

    =

    1+ 52

    1

    +

    1 52

    1

    1 = (

    1+ 52

    ) + (1 5

    2)

    = 1 = 5

    =

    Also = 1

    5

    = 15

    1

    0

    = 1

    5

    1+ 52

    1

    1

    5

    1 52

    1

    Ist .A x =

    x An

    x = n

    x

    Fn+1Fn

    =

    1 1

    1 0

    n

    1

    0

    =

    1 1

    1 0

    n

    ( 15

    1+ 52

    1

    1

    5

    1 52

    1

    ) =

  • 1 1

    1 0

    n

    ( 15

    1+ 52

    1

    1

    5

    1 52

    1

    ) = 1

    5

    1 1

    1 0

    n

    1+ 52

    1

    1

    5

    1 1

    1 0

    n

    1 52

    1

    = 15

    1+ 5

    2

    n

    1+ 52

    1

    1

    5

    1 5

    2

    n

    1 52

    1

    = 1

    5

    1+ 5

    2

    n+1

    1+ 5

    2

    n

    1 5

    2

    n+1

    1 5

    2

    n

    Fn+1Fn

    =

    1

    5

    1+ 5

    2

    n+1

    1+ 5

    2

    n

    1 5

    2

    n+1

    1 5

    2

    n

    (Binet) Fn =1

    5

    1+ 5

    2

    n

    1 5

    2

    n

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